#### Abstract

We prove a weak convergence theorem of the modified Mann iteration process for a uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mapping in a uniformly convex Banach space. We also introduce two kinds of new monotone hybrid methods and obtain strong convergence theorems for an infinitely countable family of uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mappings in a Hilbert space. The results improve and extend the corresponding ones announced by Kim and Xu (2006) and Nakajo and Takahashi (2003).

#### 1. Introduction

Let be a nonempty, closed, and convex subset of a real Banach space . We denote by the set of fixed points of , that is, . Then is said to be

(i)*nonexpansive*if for all ;(ii)

*asymptotically nonexpansive*if there exists a sequence , and for all and ;(iii)

*asymptotically quasi-nonexpansive*if there exists a sequence , and for all and ;(iv)

*generalized asymptotically nonexpansive*[1] if there exist nonnegative real sequences and with , and such that for all and ;(v)

*generalized asymptotically quasi-nonexpansive*[1] if there exist nonnegative real sequences and with , and such that for all , and ;(vi)

*asymptotically nonexpansive in the weak sense*[2] if for all .(vii)

*asymptotically nonexpansive in the intermediate sense*[3] if (viii)

*uniformly L-Lipschitzian*if there exists a constant such that for all and .

It is clear that a generalized asymptotically quasi-nonexpansive mapping is to unify various classes of mappings associated with the class of generalized asymptotically nonexpansive mapping, asymptotically nonexpansive mappings, and nonexpansive mappings. However, the converse of each of above statement may be not true. The example shows that a generalized asymptotically quasi-nonexpansive mapping is not an asymptotically quasi-nonexpansive mapping; see [1]. Note that if is asymptotically nonexpansive in the weak sense, we have that

for all , where so that . Hence, is a generalized asymptotically nonexpansive mapping.

The mapping is said to be *demiclosed* at if for each sequence in converging weakly to and converging strongly to , we have .

A Banach space is said to satisfy *Opial's property*, see [4], if for each and each sequence weakly convergent to , the following condition holds for all :

Let be a Hausdorff linear topology and let satisfy the uniform -Opial property. In 1993, Bruck, Kuczumow, and Reich proved that is -convergent if and only if is -asymptotically regular, that is,

Moreover, they also proved that the -limit of is a fixed point of .

In 1953, Mann [5] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping in a Hilbert space :

where the initial point is taken in arbitrarily and is a sequence in .

However, we note that Mann's iteration process (1.11) has only weak convergence, in general; for instance, see [6–8].

In 2003, Nakajo and Takahashi [9] proposed the following modification of the Mann iteration for a single nonexpansive mapping in a Hilbert space. They proved the following theorem.

Theorem 1.1. *Let be a closed and convex subset of a Hilbert space H and let be a nonexpansive mapping such that . Assume that is a sequence in such that for some . Define a sequence in C by the following algorithm:
**
Then defined by (1.12) converges strongly to .*

Recently, Kim and Xu [10] extended the result of Nakajo and Takahashi [9] from nonexpansive mappings to asymptotically nonexpansive mappings. They proved the following theorem.

Theorem 1.2. *Let C be a nonempty, bounded, closed, and convex subset of a Hilbert space H and let be an asymptotically nonexpansive mapping with a sequence such that as . Assume that is a sequence in such that . Define a sequence in C by the following algorithm:
**
where , as . Then defined by (1.13) converges strongly to .*

Since 2003, the strong convergence problems of the CQ method for fixed point iteration processes in a Hilbert space or a Banach space have been studied by many authors; see [9–20].

Let be an infinitely family of uniformly -Lipschitzian and generalized asymptotically quasi-nonexpansive mappings and let . In this paper, motivated by Kim and Xu [10] and Nakajo and Takahashi [9], we introduce two kinds of new algorithms for finding a common fixed point of a countable family of uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mappings which are modifications of the normal Mann iterative scheme. Our iterative schemes are defined as follows.

*Algorithm 1.3. *For an initial point , compute the sequence by the iterative process:
where , .

*Algorithm 1.4. *For an initial point , compute the sequence by the iterative process:
where , .

#### 2. Preliminaries

In this section, we present some useful lemmas which will be used in our main results.

Lemma 2.1 (see [21]). *Let , , and be three sequences of nonnegative numbers such that and
**
if and , then exists.*

Lemma 2.2 (see [22]). *Let , be two fixed numbers. Then a Banach space is uniformly convex if and only if there exists a continuous, strictly increasing, and convex function with such that
**
for all and where .*

Lemma 2.3. *Let be a nonempty subset of a Banach space and let be a uniformly -Lipschitzian mapping. Let be a sequence in such that and . Then .*

*Proof. *Since is uniformly -Lipschitzian, we have
It follows that .

The following lemmas give some characterizations and useful properties of the metric projection in a Hilbert space.

Let be a real Hilbert space with inner product and norm and let be a closed and convex subset of . For every point , there exists a unique nearest point in , denoted by , such that
is called the *metric projection* of onto . We know that is a nonexpansive mapping of onto .

Lemma 2.4 (see [12]). *Let be a closed and convex subset of a real Hilbert space and let be the metric projection from onto . Given and , then if and only if the following holds:
*

Lemma 2.5 (see [9]). *Let be a nonempty, closed, and convex subset of a real Hilbert space and let be the matric projection from onto . Then the following inequality holds:
*

Lemma 2.6 (see [12]). *Let be a real Hilbert space. Then the following equations hold: *(i)*, for all ;*(ii)*, for all and .*

Lemma 2.7 (see [10]). *Let be a nonempty, closed, and convex subset of a real Hilbert space . Given and also given , the set
**
is convex and closed.*

Lemma 2.8. *Let be a closed and convex subset of a real Hilbert space . Let be a uniformly -Lipschitzian and generalized asymptotically quasi-nonexpansive mapping with nonnegative real sequences , such that , and . Then is a closed and convex subset of .*

*Proof. *Since is continuous, is closed. Next, we show that is convex. Let and . Put . By Lemma 2.6, we have
This implies that . Since is continuous, we have , so that . Hence .

#### 3. Main Results

First, we prove a weak convergence theorem for a single uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mapping in a uniformly convex Banach space.

Theorem 3.1. *Let be a uniformly convex Banach space which satisfies Opial's property. Let be a nonempty, closed, and convex subset of , and a uniformly -Lipschitzian and generalized asymptotically quasi-nonexpansive mapping with nonnegative real sequences , such that , and . Assume that is demiclosed at , where is the identity mapping of and is a sequence in such that and . Let be the sequence in generated by the modified Mann iteration process:
**
Then converges weakly to a fixed point of .*

*Proof. *Let , we have
Since , , then by Lemma 2.1 and (3.2), we obtain that
exists.

This implies that is bounded. Put . By Lemma 2.2, there is a continuous strictly increasing convex function with such that
It follows that
By our assumptions and (3.3), we get . Since is continuous strictly increasing with , we can conclude that . Observe that . It follows from Lemma 2.3 that . Since is bounded, there exists a subsequence of converging weakly to some . From and is demiclosed at , we obtain that . That is, . Next, we show that converges weakly to and take another subsequence of converging weakly to some . Again, as above, we can conclude that . Finally, we show that . Assume . Then by Opial's property of , we have
which is a contradiction. Therefore . This shows that converges weakly to .

*Remark 3.2. * In [3, Theorem ], Bruck et al. proved that if is asymptotically nonexpansive in the weak sense and is -asymptotically regular, then Picard’s iterated sequence is -convergent to a fixed point of . Since every asymptotically nonexpansive in the weak sense is a generalized asymptotically nonexpansive mapping and if its fixed point set is nonempty, then it is a generalized quasiasymptotically nonexpansive mapping. So we can apply Theorem 3.1 with a mapping which is asymptotically nonexpansive in the weak sense when its fixed point set is nonempty and obtain that the sequence generated by the modified Mann iteration process
converges weakly to a fixed point of without asymptotically regularity condition of .

In [3, Theorem ], Bruck et al. showed that if is a bounded convex subset of a uniformly convex Banach space and is sequentially -compact, is asymptotically nonexpansive in the intermediate sense and where for some sequence of nonnegative integers , then the sequence generated by
is -convergent to a fixed point of . Note that every asymptotically nonexpansive mapping in the intermediate sense is a generalized asymptotically nonexpansive mapping. Hence, Theorem 3.1 can be applied to the class of asymptotically nonexpansive mappings in the intermediate sense to obtain weak convergence of the sequence generated by
without the boundedness and compactness conditions on .

Note that the modified Mann's iteration in Theorem 3.1 has only weak convergence.

*Question 1. *How can we modify the modified Mann's iteration in order to obtain strong convergence?

In the following theorem, we introduce a monotone hybrid method with the modified Mann's iteration to obtain a strong convergence theorem for an infinite family of uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mappings.

Theorem 3.3. *Let be a closed and convex subset of a real Hilbert space . Let be an infinitely countable family of uniformly -Lipschitzian and generalized asymptotically quasi-nonexpansive mappings of into itself with nonnegative real sequences , such that , , , for all . Assume that and the sequence , for all . Then the sequence generated by Algorithm 1.3 converges strongly to .*

*Proof. *We split the proof into six steps.*Step 1. *Show that is well defined for every .

By Lemma 2.8, we obtain that is a closed and convex subset of for every . Hence, is a nonempty, closed, and convex subset of ; consequently, is well defined for every .*Step 2. *Show that is well defined.

From the definition of and , it is obvious that is closed and convex for each . It follows from Lemma 2.7 that is closed and convex for all . This implies that is closed and convex for each . Next, we will show that , for all . First, we prove that for all and . Since is a generalized asymptotically quasi-nonexpansive mapping for all , we have that for any ,
Hence, for all and . This proves that for all and . Hence, for all .

As shown in Marino and Xu [12], by induction, we can show that for all . Hence , for all , and so is well defined.*Step 3. *Show that exists.

From and , we have
On the other hand, as , we obtain
So we have that the sequence is bounded and nondecreasing. Therefore exists.*Step 4. *Show that , where .

For , by the definition of , we see that . Noting that and , by Lemma 2.5, we conclude that
It follows from Step 3 that is a Cauchy. So, we can assume that as for some . In particular, we have that
*Step 5. *Show that , for all .

Let . Since , it follows from (3.10) that
Moreover, by Lemma 2.6, we have
It follows from (3.15) and (3.16) that
Since as for all , we have by (3.14) and (3.17) that
This implies that
for all . By Lemma 2.3 and (3.14), we get that for all .*Step 6. *Show that .

Since and for all , we have for all . Hence . By Lemma 2.4, we obtain
for all . Since , we have
for all . Again by Lemma 2.4, we obtain that . This completes the proof.

Theorem 3.4. *Let be a closed and convex subset of a real Hilbert space . Let be an infinitely countable family of uniformly -Lipschitzian and generalized asymptotically quasi-nonexpansive mappings of into itself with nonnegative real sequences , such that , , , for all . Assume that and the sequence , for all . Then the sequence generated by Algorithm 1.4 converges strongly to .*

*Proof. *We divide our proof into four steps.*Step 1. *Show that is closed and convex and for all .

It follows from Lemma 2.7 that is closed and convex for all . This implies that is closed and convex for each . Next, we will show that for all . For , . Assume that for . It follows from (3.10) and the definition of that .*Step 2. *Show that exists.

From , and , for all , we have
On the other hand, as , we obtain
So we have that the sequence is bounded and nondecreasing. Therefore exists.*Step 3. *Show that , where .

For , by the definition of , we see that . By Lemma 2.5, we obtain that
From Step 2, we obtain that is Cauchy. Hence as for some and . By using the same proof as in Step 5 of Theorem 3.3, we can show that , for all .*Step 4. *Show that .

Since and , for all , we have , for all . Hence . Since , by Lemma 2.4, we have
for all , and hence,
for all . This shows that , which completes the proof.

Since a generalized asymptotically quasi-nonexpansive mapping is to unify various classes of mappings associated with the class of generalized asymptotically nonexpansive mapping, we have the following.

Corollary 3.5. *Let be a closed and convex subset of a real Hilbert space . Let be an infinitely countable family of uniformly -Lipschitzian and generalized asymptotically nonexpansive mappings of into itself with nonnegative real sequences , such that , , , for all . Assume that and the sequence , for all . Let a sequence be generated by the following manner:
**
where , . Then the sequence converges strongly to .*

Corollary 3.6. *Let be a closed and convex subset of a real Hilbert space . Let be an infinitely countable family of uniformly -Lipschitzian and generalized asymptotically nonexpansive mappings of into itself with nonnegative real sequences , such that , , , for all . Assume that and the sequence , for all . Let a sequence be generated by the following manner:
**
where , . Then the sequence converges strongly to .*

*Remark 3.7. *(i) Corollaries 3.5 and 3.6 improve and extend the main result in [10].

(ii) If we take , and for all where is a nonexpansive mapping, then Corollary 3.5 reduces to [9, Theorem 3.4].

#### Acknowledgments

The authors would like to thank the referees for valuable suggestions on the manuscript. The authors were supported by the Center of Excellence in Mathematics, the Thailand Research Fund, the Commission on Higher Education, and the Graduate School of Chiang Mai University for the financial support.